Calculus Examples

$lim_{x \to 11} \frac{3 x - 33}{\sqrt{11 x - 11}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $11$ .

$\frac{lim_{x \to 11} 3 x - 33}{lim_{x \to 11} \sqrt{11 x - 11}}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $11$ .

$\frac{lim_{x \to 11} 3 x - lim_{x \to 11} 33}{lim_{x \to 11} \sqrt{11 x - 11}}$

Step 3

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 lim_{x \to 11} x - lim_{x \to 11} 33}{lim_{x \to 11} \sqrt{11 x - 11}}$

Step 4

Evaluate the limit of $33$ which is constant as $x$ approaches $11$ .

$\frac{3 lim_{x \to 11} x - 1 \cdot 33}{lim_{x \to 11} \sqrt{11 x - 11}}$

Step 5

Move the limit under the radical sign.

$\frac{3 lim_{x \to 11} x - 1 \cdot 33}{\sqrt{lim_{x \to 11} 11 x - 11}}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $11$ .

$\frac{3 lim_{x \to 11} x - 1 \cdot 33}{\sqrt{lim_{x \to 11} 11 x - lim_{x \to 11} 11}}$

Step 7

Move the term $11$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 lim_{x \to 11} x - 1 \cdot 33}{\sqrt{11 lim_{x \to 11} x - lim_{x \to 11} 11}}$

Step 8

Evaluate the limit of $11$ which is constant as $x$ approaches $11$ .

$\frac{3 lim_{x \to 11} x - 1 \cdot 33}{\sqrt{11 lim_{x \to 11} x - 1 \cdot 11}}$

Step 9

Evaluate the limits by plugging in $11$ for all occurrences of $x$ .

Tap for more steps...

Step 9.1

Evaluate the limit of $x$ by plugging in $11$ for $x$ .

$\frac{3 \cdot 11 - 1 \cdot 33}{\sqrt{11 lim_{x \to 11} x - 1 \cdot 11}}$

Step 9.2

Evaluate the limit of $x$ by plugging in $11$ for $x$ .

$\frac{3 \cdot 11 - 1 \cdot 33}{\sqrt{11 \cdot 11 - 1 \cdot 11}}$

Step 10

Simplify the answer.

Tap for more steps...

Step 10.1

Simplify the numerator.

Tap for more steps...

Step 10.1.1

Multiply $3$ by $11$ .

$\frac{33 - 1 \cdot 33}{\sqrt{11 \cdot 11 - 1 \cdot 11}}$

Step 10.1.2

Multiply $- 1$ by $33$ .

$\frac{33 - 33}{\sqrt{11 \cdot 11 - 1 \cdot 11}}$

Step 10.1.3

Subtract $33$ from $33$ .

$\frac{0}{\sqrt{11 \cdot 11 - 1 \cdot 11}}$

Step 10.2

Simplify the denominator.

Tap for more steps...

Step 10.2.1

Multiply $11$ by $11$ .

$\frac{0}{\sqrt{121 - 1 \cdot 11}}$

Step 10.2.2

Multiply $- 1$ by $11$ .

$\frac{0}{\sqrt{121 - 11}}$

Step 10.2.3

Subtract $11$ from $121$ .

$\frac{0}{\sqrt{110}}$

Step 10.3

Divide $0$ by $\sqrt{110}$ .

$0$